In this work we prove that the problem of the embedding
of single cell quad-equations possessing the
consistency around the cube is crucial.
As already discussed in [1]} there might
exist multiple embeddings preserving on the whole
lattice the consistency around the cube.
To understand how many and what they are is then a relevant task.
We conjecture that in the procedure of extension might
be essential to form bi-quadratic patterns
[2]}, [3]}, [4]}.
To form a bi-quadratic pattern means that, given a quad-equation
on the single cell (REF ), to construct an equation
on the \(\mathbb {Z}^{2}\)  lattice in a way such that
the bi-quadratics (REF ) pertaining to the sides
of neighboring cells are of the same type.
However preserving the bi-quadratic patterns might be ineffectual
in the case of the \(H^{6}\)  due to the lack of a well-defined
“black/white” assignment to the vertexes of the elementary cell.
